Crit\`ere d'irr\'eductibilit\'e pour les courbes elliptiques semi-stables sur un corps de nombres

TL;DR

This paper establishes explicit bounds on primes for which the Galois representation of semi-stable elliptic curves over a fixed number field is reducible, generalizing prior results and depending only on the number field.

Contribution

It provides a new explicit bound for reducible primes of semi-stable elliptic curves over number fields, extending Kraus's earlier work.

Findings

Explicit bounds depend only on the number field

Generalizes Kraus's previous results

Applicable to semi-stable elliptic curves over fixed number fields

Abstract

For a fixed number field and an elliptic curve defined and semi-stable over this number field, we consider the set of prime numbers p such that the Galois representation attached to the p-torsion points of the elliptic curve is reducible. When the number field satisfies a certain necessary condition, we give an explicit bound, depending only on the number field and not on the semi-stable elliptic curve, for these primes. This generalizes previous results of Kraus.